Calculus with Geogebra
Area under and between curves
Geogebra is capable of doing Riemann sums for you. This is helpful with visualizing the rectangles as well as checking your work.
In the input bar, type y = sin(x)
Now to estimate the area between the curve and the x-axis between x = 0 and x = , with 8 left-hand rectangles, type
LeftSum(sinx, 0, pi, 8).
You know that in order to find the actual area between the curve and the x-axis, you need to compute the following,
To use Geogebra to calculate this integral, type Integral(sinx,0,pi).
Now graph cos(x) in the same window.
To have Geogebra, calculate the area between the cosine and sine curves from 0 to type
IntegralBetween(cosx, sinx, 0, pi/4).
Riemann Sum
Write out the Riemann Sum, using left-hand endpoints, that corresponds with the calculation that was done with Geogebra. You will need to use Math mode to type out the expression.
Integral for area between curves
Type out the integral that would be needed to calculate the area between the cosine and sine curve between 0 and pi/4. You will need to use Math mode to type out the integral.
Length of parametric curves
In Calculus II, you learned about parametric curves and how to find the length of a parametric curve using the following integral:
Let's graph the following parametric curve and find its length using Geogebra.
To graph this, type Curve(t+cos(t),t-sin(t),t,0,2 π)
Once you type that in, it will assign a variable to your curve, possibly a. To find the length of the curve, type
Length(a,0,2pi).
Length of curve integral
In the box below, type out the integral that you would need to solve in order to find the arc length using calculus.
Graphing a polar curve
In order to graph a polar curve in Geogebra, one must use the Curve command. If we want to graph we have to type:
Curve((2+sin(t); t),t,0,2 π)
When you do this the curve will be designated by a variable, say a. Looking at the graph, you can see where the polar curve has a horizontal tangent. Plot that point on the curve.
From calculus we know that a horizontal tangent occurs when the derivative is zero. Now graph the derivative of the polar curve, by typing:
a'(t), using the single quotation mark for the prime symbol.
Graphing polar coordinates is one example where Desmos is much friendlier than Geogebra. Go to desmos.com, and click on graphing calculator.
Click on the wrench in the upper right corner of the screen and under Grid, click on the polar grid.
Now in the input bar, type r = 2 +sin(theta)
However, it doesn't seem to graph the derivative as easily as Geogebra.
Point of horizontal tangent
What value of the variable t corresponds to the point where the tangent line is horizontal?
Intersection of two polar graphs
Reminder that Geogebra will give your graphs names when you type them in as curves. Let's graph the folllowing polar graphs and find the point(s) of intersection.
Type the following into the input bar in Geogebra.
Curve((sqrt(3) cos(t); t),t,0,2 π)
Curve((sin(t); t),t,0,2 π)
After looking at the graphs, you can see that they intersect at two points. Let's see if Geogebra can help us find those points. Type
Intersect(a,b)
First of all you will notice that it only gives coordinates for one point, second of all, you will notice that the point is given to you in Cartesian coordinates.
Finding the points of intersection in terms of the angles
Using algebra and your knowledge of the unit circle, find the t-values (angles), where the two curves intersect. Type your answers in the window below.